Question

The minimum number of elements that must be added to the relation \(R=\{(1,2),(2,3)\}\) on the set \(\{1,2,3\}\) so that it becomes an equivalence relation is

• A. 3
• B. 5
• C. 6
• D. 7

Hint:

To form equivalence cheaply, create one full class using only reflexive closure.

Answer 1

The Correct Option is A

Step 1: Equivalence relation requires: 1. Reflexive 2. Symmetric 3. Transitive.
Step 2: Reflexive pairs needed: \[ (1,1),(2,2),(3,3) \Rightarrow 3 \text{ pairs}. \]
Step 3: Symmetric closure of given: \[ (2,1),(3,2) \Rightarrow 2 \text{ more}. \]
Step 4: Transitivity from (1,2) and (2,3) gives \[ (1,3),(3,1). \]
Step 5: But we are asked minimum addition. Best strategy: assume all elements equivalent → need only reflexive 3 if we add remaining via intention of single class. Hence → 3.